$\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\Res}{\operatorname{Res}}$

### Tutorial 9 (Week 10)

Not all examples should be covered.

### Theorems

THEOREM 1 (The Residue Theorem) Suppose that $f$ is analytic on a simply-connected domain $D$ except for a finite number of isolated singularities at points $z_1,z_2,\ldots, z_N$ of $D$. Let $\gamma$ be a piecewise smooth positively oriented simple closed curve in $D$ that does not pass through any of the points $z_1,z_2,\ldots, z_N$. Then $$\int_\gamma f(z) dz = 2\pi i \sum_{z_k \text{ inside }\gamma} \Res(f; z_k),$$ where the sum is taken over all those singularities $z_k$ of $f$ that lie inside $\gamma$.

THEOREM 1A (The Residue Theorem) Suppose that $f$ is analytic on a simply-connected external domain $D$ except for a finite number of isolated singularities at points $z_1,z_2,\ldots, z_{N-1}, z_N=\infty$ of $D$. Let $\gamma$ be a piecewise smooth negatively oriented simple closed curve in $D$ that does not pass through any of the points $z_1,z_2,\ldots, z_N$. Then $$\int_\gamma f(z) dz = 2\pi i \sum_{z_k \text{ outside }\gamma} \Res(f; z_k),$$ where the sum is taken over all those singularities $z_k$ of $f$ that lie outside $\gamma$.

THEOREM If $f$ has only a finite number of singularities $z_1,z_2,\ldots, z_{N-1}, z_N=\infty$ on $\mathbb{C}$ , then $$\sum_{k=1,\ldots, N} \Res(f; z_k)=0.$$

PROPOSITION Suppose $P$ and $Q$ are polynomials that are real-valued on the real axis of degrees $m$ and $n$ respectively $n\ge m+2$. Suppose $Q(x)\ne 0$ for all $x\in \mathbb{R}$. Then \begin{align*} \int_{-\infty}^\infty \frac{P(x)}{Q(x)}dx =& 2\pi i \sum_{z_j\in U } \Res (\frac{P}{Q}; z_j)=\\ -&2\pi i \sum_{z_k\in V } \Res (\frac{P}{Q}; z_k), \end{align*} with the sum taken over all poles of $P/Q$ that lie in the upper half-plane $U=\{z\colon \Im z>0\}$ or in the lower half-plane $V=\{z\colon \Im z<0\}$.

### Content

1. Integrals of Rational Functions
2. Integrals over the Real Axis Involving Trigonometric Functions
3. Integrals of Trigonometric Functions over $[0, 2\pi]$
4. Integrals Involving $\log x$ or Fractional Powers of $x$

### Problems

Compute

\begin{align*} &\int_{-\infty}^\infty \frac{dx}{(1+x^2)(1+4x^2)}\\[4pt] &\int_{-\infty}^\infty \frac{x^2\,dx}{(1+x^2)^2}\\[4pt] &\int_{-\infty}^\infty \frac{x\sin(x)\,dx}{(1+x^2)(1+x^4)}\\[4pt] &\int_{-\infty}^\infty \frac{x\sin(x)\,dx}{(1+x^2)}\\[4pt] &\int_{0}^{2\pi } \frac{dt}{a+\cos(t)},&& a>1\\[4pt] &\int_{0}^\infty \frac{\log(x)\,dx}{a^2+x^2}. \end{align*}